On an integral equation of Volterra type. (Russian) Zbl 0653.45005

The following integral equation is investigated (1) \(x(t)=x_ 0+\int^{t}_{-t}K(t,s,x(s))ds,\) \(t\in [-1,1]\). Assume that the function K is continuous on the set \([-1,1]\times [-1,1]\times [x_ 0- r,x_ 0+r]\) and satisfies the condition \(\lim_{t_ 2\to t_ 1}\int^{t_ 1}_{-t_ 1}| K(t_ 2,s,x)-K(t_ 1,s,x)| ds=0,\) where \(t_ 1,t_ 2\in [-1,1]\). Then the equation (1) possesses at least one solution on [-1,1] which may be obtained by Tonelli ’s procedure or by the method of successive approximation.
Reviewer: J.Banaś


45G10 Other nonlinear integral equations
45L05 Theoretical approximation of solutions to integral equations